Abstract
In this work, meshless methods based on a radial basis function (RBF) are applied for the solution of two-dimensional steady-state heat conduction problems with nonlocal multi-point boundary conditions (NMBC). These meshless procedures are based on the multiquadric (MQ) RBF and its modified version (i.e., integrated MQ RBF). The meshless method is extended to the NMBC and compared with the standard collocation method (i.e., Kansa’s method). In extended methods, the interior and the boundary solutions are approximated with a sum of RBF series, while in Kansa’s method, a single series of RBF is considered. Three different sorts of solution domains are considered in which Dirichlet or Neumann boundary conditions are specified on some part of the boundary while the unknown function values of the remaining portion of the boundary are related to a discrete set of interior points. The influences of NMBC on the accuracy and condition number of the system matrix associated with the proposed methods are investigated. The sensitivity of the shape parameter is also analyzed in the proposed methods. The performance of the proposed approaches in terms of accuracy and efficiency is confirmed on the benchmark problems.
Highlights
The partial differential equations with nonlocal boundary conditions that emerged in the literature have significant applications in the fields of engineering, astrophysics, and biology
This paper aims to investigate the effect of this parameter on the property of the radial basis function (RBF)-based method using integrated MQ RBF
We have implemented four different methods based on radial basis functions, i.e., the Kansa method and the Kansa method, and we extended the method proposed in [59] to the complicated nonlocal boundary value problems
Summary
The partial differential equations with nonlocal boundary conditions that emerged in the literature have significant applications in the fields of engineering, astrophysics, and biology. The two-dimensional diffusion equation subject to a nonlocal condition involving a double integral along with Neumann’s boundary conditions was formulated in a rectangular region Such a problem is solved numerically using a meshless collocation method [37]. In paper [38], the RBF-based collocation technique was applied to investigate the influences of nonlocal conditions on the optimal selection of shape parameter e, ill-conditioning, and accuracy of the method. It is known that the meshless method can be successfully applied to solve such nonlocal problems It was shown (e.g., see paper [11,38,40]) that due to the nonlocal parameter γ, the condition number κ of the system matrix involved and accuracy of the method are affected.
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